let-A-n-cos-2-n-x-x-2-3-2-dx-1-calculate-A-n-interms-of-n-2-find-nsture-of-the-serie-A-n-and-n-n-A-n-

Question Number 65679 by mathmax by abdo last updated on 01/Aug/19

l e t A_{n} = \int_{- \infty}^{+ \infty} \frac{c o s (2^{n} x)}{{(x^{2} + 3)}^{2}} d x

1) c a l c u l a t e A_{n} i n t e r m s o f n

2) f i n d n s t u r e o f t h e s e r i e Σ A_{n} a n d Σ n^{n} A_{n}

Commented by mathmax by abdo last updated on 02/Aug/19

2) \sum_{n = 0}^{\infty} A_{n} = \frac{π}{6} \sum_{n = 0}^{\infty} 2^{n} e^{- \sqrt{3} 2^{n}} + \frac{π \sqrt{3}}{18} \sum_{n = 0}^{\infty} e^{- \sqrt{3} 2^{n}}

t h o s e s e r i e s c o n v e r g e s \Rightarrow Σ A_{n} c o n v e r g e s

w e c a n v e r i f y t h a t {l i m}_{n \to + \infty} n^{n} A_{n} \neq 0 \Rightarrow Σ A_{n} d i v e r g e s

Commented by mathmax by abdo last updated on 02/Aug/19

1) w e h a v e A_{n} = \int_{- \infty}^{+ \infty} \frac{c o s (2^{n} x)}{{(x^{2} + 3)}^{2}} d x \Rightarrow A_{n} = R e (\int_{- \infty}^{+ \infty} \frac{e^{i 2^{n} x}}{{(x^{2} + 3)}^{2}} d x)

l e t φ (z) = \frac{e^{i 2^{n} z}}{{(z^{2} + 3)}^{2}} p o l e s o f φ ?

φ (z) = \frac{e^{i 2^{n} z}}{{(z - i \sqrt{3})}^{2} {(z + i \sqrt{3})}^{2}} t h e p o l e s o f φ a r e \bar{+} i \sqrt{3} (d o u b l e s)

r e s i d u s t h e o r e m g i v e \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π R e s (φ, i \sqrt{3})

R e s (φ, i \sqrt{3}) = {l i m}_{z \to i \sqrt{3}} \frac{1}{(2 - 1)!} {{(z - i \sqrt{3})}^{2} φ (z)}^{(1)}

= {l i m}_{z \to i \sqrt{3}} {\frac{e^{i 2^{n} z}}{{(z + i \sqrt{3})}^{2}}}^{(1)} = {l i m}_{z \to i \sqrt{3}} \frac{i 2^{n} e^{i 2^{n} z} {(z + i \sqrt{3})}^{2} - 2 (z + i \sqrt{3}) e^{i 2^{n} z}}{{(z + i \sqrt{3})}^{4}}

= {l i m}_{z \to i \sqrt{3}} \frac{i 2^{n} e^{i 2^{n} z} (z + i \sqrt{3}) - 2 e^{i 2^{n} z}}{{(z + i \sqrt{3})}^{3}}

= \frac{i 2^{n} e^{i 2^{n} (i \sqrt{3})} (2 i \sqrt{3}) - 2 e^{i 2^{n} (i \sqrt{3})}}{{(2 i \sqrt{3})}^{3}} = \frac{- 2 \sqrt{3} 2^{n} e^{- 2^{n} \sqrt{3}} - 2 e^{- 2^{n} \sqrt{3}}}{(- 8 i) (3 \sqrt{3})}

= \frac{(2 \sqrt{3} 2^{n} + 2) e^{- 2^{n} \sqrt{3}}}{24 i \sqrt{3}} \Rightarrow \int_{- \infty}^{+ \infty} φ (z) d z = 2 i π \frac{(\sqrt{3} 2^{n} + 1) e^{- 2^{n} \sqrt{3}}}{12 i \sqrt{3}}

= \frac{π}{6 \sqrt{3}} (\sqrt{3} 2^{n} + 1) e^{- 2^{n} \sqrt{3}} \Rightarrow A_{n} = \frac{π}{18} {3 . 2^{n} + \sqrt{3}) e^{- \sqrt{3} 2^{n}}